Question: Heat equation with time dependant boudary condition

Hi,

I'm currently working on chemical process thermal exchange and particularly on the solving of the heat equation using a time dependant boundary condition.

Briefly, the process consists in two layers of different materials (M1 and M2, thickness L1 and L2). The bottom part of the material M1 (z=0) is cooled down from Ti to Tf with the function T(0,t)=Ti-R*t (R is the cooling rate in °C.min-1) until T(0,t)=Tf. Here the equilibrium is reached in t=(Tf-Ti)/R and the process keeps this temperature.

On the top of material M2, I have a water solution (kind of third layer, thickness Lw) and I would like to model the temperature at the surface of the water. I guess there is a delay between z=0 and z=L1+L2+Lw to reach Tf. The idea is to determine the time necessary for the water surface (z=L1+L2+Lw) to reach Tf and so consider the system in equilibrium.

In first approximation I consider a perfect contact between materials but if not I just have to add a new layer between M1 and M2.

I have two questions:

1. Does somebody have an idea to solve the 1D heat equation with a time dependant boudary condition ? Does it work on Maple13 ? I already tried few things to visualize the 3D plot (z,t,T(z,t)) but no one seems to work.

2. At some point, the water can freeze and I observe a brief increase in the temperature (I have a thermocouple in M2) before the water continues to cool down once freezed. Is it possible toi introduce this in the equation ?

Thanks in advance,

Regards.

Lisa